Thermal-optical mechanical waves of the microelongated semiconductor medium with fractional order heat time derivatives in a rotational field

Outlined here is an innovative method for characterizing a layer of microelongated semiconductor material under excitation. Fractional time derivatives of a heat equation with a rotational field are used to probe the model during photo-excitation processes. Micropolar-thermoelasticity theory, which the model implements, introduces the microelongation scalar function to characterize the processes occurring inside the microelements. When the microelongation parameters are considered following the photo-thermoelasticity theory, the model investigates the interaction scenario between optical-thermo-mechanical waves under the impact of rotation parameters. During electronic and thermoelastic deformation, the key governing equations have been reduced to dimensionless form. Laplace and Fourier's transformations are used to solve this mathematical problem. Isotropic, homogeneous, and linear microelongated semiconductor medium's general solutions to their respective fundamental fields are derived in two dimensions (2D). To get complete solutions, several measurements must be taken at the free surface of the medium. As an example of numerical modeling of the important fields, we will use the silicon (Si) material’s physicomechanical characteristics. Several comparisons were made using different values of relaxation time and rotation parameters, and the results were graphically shown.

www.nature.com/scientificreports/ In addition, as part of the recombination process, electrons leave holes behind when they go through a material, even if they are in the shape of a cloud. These situations often arise during hole diffusion processes, and in recent years thermo-optical theory has shown to be a valuable tool for describing the corresponding system of equations. Thermoelastic deformation processes for these semiconducting materials may also be described by introducing and implementing the notion of thermoelasticity into this area. The primary motivation for introducing this subject is to investigate, within the framework of photo-thermoelasticity theory, the impact of microelongation parameters during the investigation of semiconductor materials. In this study, the photo-thermoelasticity theory is used in the investigation of a microelongated semiconductor material in a rotating field. The model is formulated using the fractional-order heat conduction equation. Here, we account for the microinertia and microelements of the semiconductor medium. Using dimensionless variables, the governing equations in 2D deformation are translated into their non-dimensional versions. By using the Laplace and Fourier transforms approach, we can derive the physical domain expressions for a variety of physical variables with specific boundary conditions. The rotation field's influence on the simulated wave propagations is shown visually, along with some comparisons, concerning the micro-elongation parameters and fractional parameters.

Theoretical model and basic equations
Plasma wave propagation is described by the optical function, which is the carrier density N . The temperature variation T , which quantifies the thermal effect, may be used to illustrate the thermal distribution. It is possible to introduce the distribution of elastic waves with the help of the displacement vector u i . Lastly, the effect of elongation is described by the scalar micro-elongation function ϕ . A semiconductor medium will undergo a phase transition if a uniform rotating velocity ( = n ) is supplied along the y-axis (Fig. 1). Assuming that the material is homogeneous, isotropic, thermoelastic, and a photothermal semiconductor, the governing equations of plasma transport coupling may be written as [42][43][44][45] : According to the photo-thermoelasticity theory, the microelongated constitutive equations of semiconductors in the tensor form are 12-16 : where the "comma" before an index suggests space-differentiation, the dot" above a symbol suggests timedifferentiation, u i is the velocity of a particle and e = e II = u I,I is the volumetric strain.
It is possible to express the plasma transport (diffusion) equation in such a way that it describes the interaction between thermal waves and plasma waves as 46 : When the medium is in a state of microelongation in accordance with the processes of microelements, the motion and microinertia equations, which are valid under the influence of the rotating field, may be presented as follows 47 :  www.nature.com/scientificreports/ Following is a definition that may be presented in accordance with the Riemann-Liouville fractional integral operator 20 : The Riemann-Liouville fractional integral of order B is I B of any function X(�) ( Ŵ(B) refers to the gamma function). The Caputo fractional derivative is ∂ B ∂t B for the continuous function (Lebesgue function) X(�) which can be represented as 26-28 : On the other hand, the definition of the fractional derivative proposed by Caputo is: Elastic-electronic body theory yields the following form for the time-fractional heat conductive equation in a microelongated semiconductor medium: (the problem is studied in case of 0<α ≤ 1 , the superconductivity is obtained when 1 < B < 2) 16 : The analysis is simplified when a 2D problem is considered. In this case, the 2D deformation of the displacement vector and the microelongation scalar function can be represented in xz-plane as: Microelongation coefficient of the linear thermal expansions is ( α t 2 ), κ = ∂n 0 ∂T T τ which represents a coupling thermal activation parameter and γ 1 = (3 + 2µ)α t 2 parameter depending on microelongational semiconductor. The fundamental governing Eqs. (2)-(5) may be reformulated for 2D perturbation as 37-39 : By substituting a suitable scale, such as a characteristic length, time, or temperature, into the main equations, the dimensional (or physical) terms may be transformed into the non-dimensional ones.
By deleting the superscripts, Eq. (13) may be utilized to transform all the primary equations into the form below: d� n d�, n − 1 < B < n. www.nature.com/scientificreports/ Helmholtz's theorem allows us to express translations as functions in both scalar �(x, z, t) and vector space-time �(x, z, t) = (0, ψ, 0) , which we may write as: The above set of Eqs. (14)- (17), may be rearranged using Eq. (18) to provide the following: Recasting the 2D constitutive relations yields: , C 6 =γ 1 ρω * 2 T ô γ α 0 . Initial conditions satisfying the following homogeneous requirements may be considered in finding a solution to the problem:

Formulation in the transform domain
Using their specified Laplace and Fourier transformations for every function ζ(x, z, t) , in addition, the Laplace and Fourier transform form for the Caputo derivative, which is defined as:  (14) and (19)- (23), yields: where Equations (26)-(30) are shown to be linked differential equations from the given set of equations. The following tenth-order differential equation is fulfilled by φ,Ñ,T,� and ψ may be obtained by using the elimination approach to the system of Eqs. (26)-(30) as: The following is a possible factorization of Eq. (32): where k 2 n (n = 1, 2, 3, 4, 5 : Re(k n ) > 0) represent the roots of the auxiliary Eq. (33). General form linear solutions to Eq. (32) may be expressed in terms of their roots which are bounded x → ∞ when as: www.nature.com/scientificreports/ where n , ′ n , ′′ n , ′′′ n and ′′′′ n express arbitrary unknown constants.
The components of displacement and stress components described in Eqs. (18) and (31), in terms of nondimensional variables specified, assume the form:

Boundary conditions
At the boundary ( x = 0 ) of the fractional microelongated surface, you may choose certain boundary conditions that will determine the values of uncertain parameters n 44 . The requirements might be stated as.
The two mechanical conditions are chosen as loaded ( P ) for normal stress and freely for tangent stress using the above transformation, which yields: www.nature.com/scientificreports/ The thermal condition under the above transformation can be chosen in a thermally shocked with reference temperature Q case as: The elongation can be chosen free under the transformation at x = 0 as: According to the recombination processes in the semiconductor medium, the plasma condition can be chosen when the concentration of the electrons ñ 0 is obtained with the speed of recombination s , which can be represented in the following form: Using the expressions of T ,σ xx ,σ xz ,φ and Ñ under the transformations according to the Eqs. (41)-(44), we get: The complete solutions of the main physical quantities are obtained when the above system of according to Eq. (45), which are solved using the inverse of matrix technique to obtain the unknown parameters n .

Inversion of the Laplace-Fourier transforms
The inversion of the above main equations in the time physical domain is required to get the full solutions of the 2D distributions of dimensionless physical field variables. For problems in two dimensions in Cartesian coordinates, this is the generic solution in the domain of the Laplace-Fourier transform.
It is possible to express the inverse Fourier transform as: Nonetheless, a Riemann-sum approximation approach is employed for the numerical inversion of Laplace transforms 36 .
The inverse of a function ζ (x, z, s) in the Laplace domain may be rewritten as: where n represents a greater arbitrary constant than all real parts of the singularities of ζ (x, z, s) , s = n + iM (n, M ∈ R ), on the other hand, the inverted of Eq. (47) can be represented as: The Fourier series expansion can be used for the function ζ(x, z, t ′ ) in the closed interval 0, 2t ′ , which yields: where Re represent the real section and i = √ −1 . The sufficient N is a large integer that can be chosen freely 36 .

Discussion and numerical results
We now carry out some numerical calculations in order to further analyze the problem and determine how different characteristics such as rotation parameter, fractional parameter, and phase lag times included in the medium affect the physical fields. Input parameters for a fractional microelongated semiconductor material such as silicon (Si) are used to run numerical simulations. The numerical findings may be visually shown in MATLAB (2022a). The Si parameters needed to create a graphical simulation using the SI unit of the physically relevant constants are 45-51 : In this work, we calculate the wave distributions of the principal fields in 2D using non-dimensional variables. Small-time numerical simulations are performed in the 0 ≤ x ≤ 10 range.
Impact of thermal memories. The influence of relaxation time on the variation of basic fractional physical variables as a function of horizontal distance ( 0 ≤ x ≤ 10 ) is shown in Fig. 2. According to the various models in photo-thermoelasticity theory, the relaxation times are selected in this situation (three models: CD, LS and DPL). When, six different types of wave propagation are depicted: thermal (temperature distributions), microelongation, elastic (displacement), plasma (carrier intensity), and mechanical (stresses σ xx and σ xz ). The free surface of the excited fractional microelongated semiconductor is shown in Fig. 2 to have physical distributions that conform to the boundary conditions when t = 0.001 , B = 0.0 and = 0.3 . Light's thermal loads cause a thermal wave distribution to begin at the positive value at the surface and grow until they reach their maximum www.nature.com/scientificreports/ value in the first range. In the second range, for both the CD and LS models, the thermal wave gradually drops to its lowest value at the zero line, but for the DPL model, the thermal wave first grows and then gradually decreases to its minimum value at the zero line. For CD and LS models, however, the wave distributions of plasma and elastic (displacement) waves follow the same pattern as the thermal wave distribution. The DPL model's distribution, on the other hand, exhibits the same behavior as the CD model (exponential behavior), however, the magnitude varies with the values of thermal relaxation durations. All three sets of numerical findings (temperature, displacement, and carrier density) are in agreement with the experimental data 52 . For three distinct photothermoelasticity models, the distribution of microelongation vibration against distance is shown in the second inset figure. Three different instances of the thermal relaxation time (CD, LS and DPL) are shown in Fig. 2 as a www.nature.com/scientificreports/ fluctuating field of microelongation changes with increasing horizontal distance. Microelongation starts at zero at the free surface and declines monotonously to its lowest, as seen in this picture, before gradually rising and reducing periodically until it again reaches zero (equilibrium state). It is observed that the factor of relaxation times has a substantial impact on the behavior of the microelongation function under different conditions of microelongation. As can be seen in the Figure, as the relaxation duration increases, so does the amplitude of the microelongation field. When TE and ED deformations occur, the mechanical wave's (normal stress's) surface-todepth gradient begins at negative and rapidly declines to its minimal peak value. When we go farther from the surface, the waves' propagation pattern starts to rise gradually, peaks at its greatest value, then fluctuates between a minimum and maximum values a few times before disappearing altogether. Yet, because of the thermal influence of light, the tangent stress distribution first rises before plateauing at the free surface. In contrast, in the www.nature.com/scientificreports/ second band, the wave behavior is waveform, with the wave's propagation decreasing until it coincides with the zero line, and then disappearing entirely when the system reaches equilibrium.
Impact of fraction parameter. In this section (Fig. 3), a comparison study illustrates the influence of  www.nature.com/scientificreports/ ures, the boundary conditions for all of the physical quantities are satisfied, and all of the curves coincide as the variable x approaches infinity. The wave propagation of the primary physical variables saw a rise in amplitude in response to an increase in the fractional time derivative parameter.
Impact of rotation parameter. In two instances in the range 0 ≤ x ≤ 10 , Fig. 4 (composed of six subfigures) depicts how the propagation of thermal, microelongation elastic, plasma, and mechanical waves (and) change for constant values of dimensionless time t = 0.001 . According to the DPL model, there are two possible scenarios: one in which the medium is studied while under the influence of a rotation effect ( = 0.3 ), and another in which it is studied independently of any such effect ( = 0.0 ). All the wave propagations of the considered fields are shown to be significantly affected by the rotation field parameter in this figure.

Conclusion
With a set of input physical parameters, an analytical formulation is offered and visually shown for a rotation field with fractional order to heat equation acting on an isotropic-homogeneous-microelongated semiconducting elastic material. According to the generalized photo-thermoelasticity theory, the main equations in 2D are established, which describe the interplay between thermal, mechanical, microelongation, and carrier intensity. The photo-excitation transport mechanisms in the microelongated semiconductor material are investigated. According to the various types of thermal memory, three models of the photo-thermoelasticity theory are considered (CD, LS, and DPL). Microelongated silicon semiconducting media are simulated numerically under controlled circumstances. All physical distributions of waves in propagation have been shown to eventually settle into a stable equilibrium. All physical quantities tend to vary more consistently. It was also discovered that the wave propagation of the physical variables under examination is significantly affected by the relaxation times. Compared to the LS, and CD theories of photo-thermoelasticity, DPL has been shown to have superior vibrational behavior. The results of the thermoelastic heat equation are significantly impacted by the existence of the fractional time derivative. Moreover, the propagating waves show an obvious influence on the rotation parameter. Microelongated semiconductor silicon is very important to research and has several potential applications in today's state-of-the-art electronic gadgets, including but not limited to sensors, computer processors, diodes, accelerometers, inertial sensors, and electric circuits.

Data availability
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